Question: $ A = \left[\begin{array}{rrr}0 & 1 & 4 \\ 2 & 3 & -1\end{array}\right]$ $ F = \left[\begin{array}{rr}5 & -2 \\ 1 & 0 \\ 0 & 0\end{array}\right]$ What is $ A F$ ?
Solution: Because $ A$ has dimensions $(2\times3)$ and $ F$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ A F = \left[\begin{array}{rrr}{0} & {1} & {4} \\ {2} & {3} & {-1}\end{array}\right] \left[\begin{array}{rr}{5} & \color{#DF0030}{-2} \\ {1} & \color{#DF0030}{0} \\ {0} & \color{#DF0030}{0}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rr}{0}\cdot{5}+{1}\cdot{1}+{4}\cdot{0} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{0}\cdot{5}+{1}\cdot{1}+{4}\cdot{0} & ? \\ {2}\cdot{5}+{3}\cdot{1}+{-1}\cdot{0} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{0}\cdot{5}+{1}\cdot{1}+{4}\cdot{0} & {0}\cdot\color{#DF0030}{-2}+{1}\cdot\color{#DF0030}{0}+{4}\cdot\color{#DF0030}{0} \\ {2}\cdot{5}+{3}\cdot{1}+{-1}\cdot{0} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{0}\cdot{5}+{1}\cdot{1}+{4}\cdot{0} & {0}\cdot\color{#DF0030}{-2}+{1}\cdot\color{#DF0030}{0}+{4}\cdot\color{#DF0030}{0} \\ {2}\cdot{5}+{3}\cdot{1}+{-1}\cdot{0} & {2}\cdot\color{#DF0030}{-2}+{3}\cdot\color{#DF0030}{0}+{-1}\cdot\color{#DF0030}{0}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}1 & 0 \\ 13 & -4\end{array}\right] $